Coxeter groups, symmetries, and rooted representations

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Olivier Geneste, Luis Paris

To cite this version:

Olivier Geneste, Luis Paris. Coxeter groups, symmetries, and rooted representations. Communications

in Algebra, Taylor & Francis, 2018, 46 (5), pp.1996-2002. �10.1080/00927872.2017.1363221�. �hal-

01404050�

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Coxeter groups, symmetries, and rooted representations

O

LIVIER

G

ENESTE

L

UIS

P

ARIS

20F55

1 Introduction

Let Γ be a Coxeter graph, and let (W, S) be the Coxeter system of Γ . A symmetry of Γ is a permutation g of S such that m

_g(s),g(t)

= m

_s,t

for all s, t ∈ S , where (m

s,t

)

s,t∈S

is the Coxeter matrix of Γ . Let G be a group of symmetries of Γ. Then G is necessarily finite, and it can be viewed as a group of automorphisms of W . We denote by W

^G

the subgroup of W fixed under the action of G. M¨uhlherr [7] and H´ee [4], independently of one another, proved that W

^G

is a Coxeter group. None of them gave explicitly the Coxeter graph ˜ Γ which defines W

^G

. However, a third proof, different from the other two, with an explicit description of ˜ Γ, is given in Crisp [2,

3].

Let Π = {

_s

| s ∈ S} be a set in one-to-one correspondence with S , let V be the real vector space having Π as a basis, and let h., .i be the symmetric bilinear form on V defined by h

_s

,

t

i = − cos(π/m

s,t

) if m

_s,t

6= ∞ and h

_s

,

t

i = −1 if m

_s,t

= ∞. For every s ∈ S we define the linear transformation f

s

: V → V by f

s

(x) = x − 2 hx,

_s

i

_s

. Then the map S → GL(V ), s 7→ f

s

, induces a celebrated faithful linear representation f : W → GL(V), called the canonical representation of W (see Bourbaki [1]). In our context, the triple (V, h., .i, Π) will be called the canonical root basis of Γ .

Let G be a group of symmetries of Γ. Then G acts on V sending

s

to

g(s)

for all s ∈ S , and this action leaves invariant the canonical form h., .i. Hence, the canonical representation f : W → GL(V) is equivariant, in the sense that f (g(w)) = g◦ f (w) ◦ g

⁻¹

for all g ∈ G and all w ∈ W , and therefore f induces a linear representation f

^G

: W

^G

→ GL(V

^G

), where V

^G

= {x ∈ V | g(x) = x for all g ∈ G}.

A naive question would be: Is f

^G

: W

^G

→ GL(V

^G

) the canonical representation of

W

^G

? A positive answer would provide a way to (re)prove that W

^G

is a Coxeter group

and to determine the Coxeter graph of W

^G

. Unfortunately, simple calculations show

that f

^G

is not the canonical representation in general. Nevertheless, one can transpose

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this question to a larger family of linear representations, the rooted representations introduced by Krammer [5,

6], and, in this context, the answer is yes. Our purpose is

to show that.

A root basis of Γ is a triple (V, h., .i, Π), where V is a finite dimensional real vector space, h., .i is a symmetric bilinear form on V , and Π = {

s

| s ∈ S} is a collection of vectors in V in one-to-one correspondence with S , that satisfies the following properties:

(a) h

_s

,

s

i = 1 for all s ∈ S ; (b) for all s, t ∈ S, s 6= t, we have

h

_s

,

t

i = − cos(π/m

s,t

) if m

_s,t

6= ∞ , h

_s

,

t

i ∈ (−∞, −1] if m

_s,t

= ∞ ; (c) there exists χ ∈ V

^∗

such that χ(

s

) > 0 for all s ∈ S.

As mentioned above, if Π is a basis of V and h

_s

,

t

i = −1 whenever m

_s,t

= ∞ , then (V, h., .i, Π) is called the canonical root basis of Γ .

This definition is taken from Krammer’s thesis [5,

6]. It is both, a generalization of the

canonical spaces and canonical forms defined by Bourbaki [1], and a new point of view on the theory of reflection groups developed by Vinberg [8]. Note also that Condition (c) in the above definition often follows from Conditions (a) and (b), but not always (see Krammer [6, Proposition 6.1.2]).

Let (V, h., .i, Π) be a root basis of Γ . For every s ∈ S we define the linear transformation f

s

: V → V by f

s

(x) = x − 2hx,

_s

i

_s

. The following theorem can be proved for any root basis in the same way as it is proved in Bourbaki [1] for the canonical root basis.

Theorem 1.1

(Krammer [5,

6])

The map S → GL(V) , s 7→ f

_s

, induces a faithful linear representation f : W → GL(V) .

The representation f : W → GL(V) of Theorem

1.1

is called the rooted representation of W associated with (V, h., .i, Π).

Let G be a group of symmetries of Γ, and let (V, h., .i, Π) be a root basis. As for the

canonical root basis, we assume that G embeds in GL(V), satisfies g(

s

) =

g(s)

for

all g ∈ G and all s ∈ S, and leaves invariant the form h., .i . Then the representation

f : W → GL(V) is equivariant in the sense that f (g(w)) = g ◦ f (w) ◦ g

⁻¹

for all g ∈ G

and all w ∈ W , and therefore f induces a linear representation f

^G

: W

^G

→ GL(V

^G

),

where V

^G

= {x ∈ V | g(x) = x for all g ∈ G}. The goal of this paper is to prove the

following.

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Coxeter groups, symmetries, and rooted representations 3

Theorem 1.2

(1) The group W

^G

is a Coxeter group.

(2) Let Γ ˜ denote the Coxeter graph of W

^G

. There exists a subset Π ˜ of V

^G

such that (V

^G

, h., .i, Π) ˜ is a root basis of Γ, and the induced representation ˜ f

^G

: W

^G

→ GL(V

^G

) is the rooted representation associated with (V

^G

, h., .i, Π) ˜ . In particular, f

^G

is faithful.

A similar approach is adopted in H´ee [4, Section 3], with different definitions. One can easily show that the root system obtained from a root basis is a root system in Hée sense [4], and that part of the results of the paper, such as the fact that (V

^G

, h., .i, Π) is a ˜ root basis, can be deduced from Hée [4]. However, to get the explicit expression of the Coxeter graph ˜ Γ of W

^G

, one would need extra arguments that can be either a rewrite of Lemma

3.3, or some arguments similar to that given in Crisp [2,3]. More generally,

the whole theorem is more or less in the literature. In particular, as mentioned before, Part (1) is explicit in M¨uhlherr [7], H´ee [4] and Crisp [2,

3]. But, our aim is to provide

a new point of view on the question with unified, short and self-contained proofs.

A more precise statement of Theorem

1.2

is given in Section

2. In particular, the

Coxeter graph ˜ Γ and the set ˜ Π are explicitly described. Section

3

is dedicated to the proofs.

2 Statement

The length of an element w ∈ W , denoted by lg(w), is the shortest length of an expression of w over the elements of S . An expression w = s

₁

· · · s

_`

is called reduced if ` = lg(w). It is known that, if W is finite, then W has a unique longest element, that is, an element w

₀

∈ W such that lg(w) ≤ lg(w

0

) for all w ∈ W , and this element is an involution (see Bourbaki [1]).

For X ⊂ S, we denote by Γ

X

the full subgraph of Γ spanned by X , and by W

X

the subgroup of W generated by X. The subgroup W

_X

is called a standard parabolic subgroup of W . By Bourbaki [1], (W

X

, X) is a Coxeter system of Γ

X

. If W

X

is finite, then we denote by w

X

the longest element of W

X

.

Let G be a group of symmetries of Γ. Now, we define a Coxeter matrix ˜ M = M ˜

^G

= ( ˜ m

_X,Y

)

X,Y∈S

(and its associated Coxeter graph, ˜ Γ). This will be the Coxeter matrix (and the Coxeter graph) of W

^G

(see Theorem

1.2

and Theorem

2.2).

We denote by O the set of orbits of G in S , and we set S = {X ∈ O | W

X

is finite}.

Then S is the set of indices of ˜ M (which is the set of vertices of ˜ Γ ). Let X, Y ∈ S .

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•

If m

_s,t

= 2 for all s ∈ X and all t ∈ Y , then we set ˜ m

_X,Y

= m ˜

_Y,X

= 2.

•

Let k ∈ { 1, 2, 3, 4, 5 } . If Γ

X∪Y

is a disjoint union of copies of the Coxeter graph depicted in Figure

2.1

(k), where the vertices corresponding to x

₁

, x

₂

, . . . belong to X and the vertices corresponding to y

₁

, y

₂

, . . . belong to Y , then we set

˜

m

_X,Y

= m ˜

_Y,X

= m if k = 1, ˜ m

_X,Y

= m ˜

_Y,X

= 4 if k ∈ { 2, 3 } , ˜ m

_X,Y

= m ˜

_Y,X

= 8 if k = 4, and ˜ m

_X,Y

= m ˜

_Y,X

= 6 if k = 5. In this case we say that (X, Y ) is a bi-orbit of type k.

•

We set ˜ m

_X,Y

= m ˜

_Y,X

= ∞ in the remaining cases.

x1m y1 x1 y1 x2 x1 y1 y2 x₂

(1) (2) (3)

x₁ y1 4 y₂ x2

x1

x₂

x3

y₁

(4) (5)

Figure 2.1: Bi-orbits.

The next lemma will be used in the definition of the set ˜ Π. It is well-known and can be easily proved using [1, Chapter V, Section 4, Subsection 8].

Lemma 2.1

Let (V, h., .i, Π) be a root basis of Γ . Suppose that W is finite and that Π spans V . Then h., .i is a scalar product, and (V, h., .i, Π) is the canonical root basis of Γ. In particular, Π is a basis of V .

We turn back to the hypothesis of Theorem

1.2, that is,

Γ is any Coxeter graph, G is a group of symmetries of Γ, and (V, h., .i, Π) is a root basis of Γ . We assume that G embeds in GL(V) so that the form h., .i is invariant under the action of G, and g(

s

) =

_g(s)

for all s ∈ S and all g ∈ G.

Let X be an element of S , that is, an orbit of G in S such that W

X

is finite. Set Π

X

= {

s

| s ∈ X}, and denote by V

X

the linear subspace of V spanned by Π

X

, and by h., .i

_X

the restriction of h., .i to V

_X

× V

_X

. By Lemma

2.1,

Π

X

is a basis of V

_X

and h., .i

_X

is a scalar product. Let a

X

=

P

s∈X

s

. Note that a

X

∈ V

^G

, hence, by the above, a

_X

6= 0 and ka

_X

k > 0. We set ˜

X

=

_ka^a^X

Xk

for all X ∈ S , and ˜ Π = Π ˜

^G

= { ˜

X

| X ∈ S}.

The main result of the paper, with a precise statement, is the following.

Theorem 2.2

(1) The set S

_W

= {w

_X

| X ∈ S} generates W

^G

, and (W

^G

, S

_W

) is a

Coxeter system of Γ ˜ .

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(2) The triple (V

^G

, h., .i, Π) ˜ is a root basis of Γ, and the induced representation ˜ f

^G

: W

^G

→ GL(V

^G

) is the rooted representation associated with (V

^G

, h., .i, Π). ˜ In particular, f

^G

is faithful.

Remark

The proof of Part (1) of Theorem

2.2

uses the induced representation f

^G

: W

^G

→ GL(V

^G

). Nevertheless, the conclusion of Part (1) is always true because there is always a root basis which satisfies the hypothesis of the theorem: the canonical root basis.

3 Proof

We assume given a Coxeter graph Γ, a root basis (V, h., .i, Π) of Γ , and a group G of symmetries of Γ . We assume that G embeds in GL(V), satisfies g(

s

) =

_g(s)

for all g ∈ G and all s ∈ S, and leaves invariant the form h., .i .

Let f : W → GL(V) be the rooted representation of W associated with (V , h., .i, Π).

From now on, in order to simplify the notations, we will assume that W acts on V via f , and we will write w(x) in place of f (w)(x) for w ∈ W and x ∈ V . Lemmas

3.1

to

3.4

are preliminaries to the proof of Theorem

2.2. Lemma3.1

(1) is well-known. It is a direct consequence of M¨uhlherr [7, Lemma 2.8], and its proof can be found in the beginning of the proof of M¨uhlherr [7, Theorem 1.3]. Lemma

3.1

(2) is also know. Its proof is implicit in Crisp [2], but, as far as we know, it is not explicitly given anywhere else.

Lemma 3.1

(1) The group W

^G

is generated by S

W

.

(2) We have (w

X

w

_Y

)

^m^˜^X,Y

= 1 for all X, Y ∈ S such that m ˜

_X,Y

6= ∞ .

Proof

As mentioned above, the proof of Part (1) can be found in M¨uhlherr [7]. So, we only need to prove Part (2). Let X ⊂ S be such that Γ

X

is a disjoint union of vertices (i.e. Γ

X

has no edge). Then W

X

is finite and w

X

=

Q

s∈X

s. Let X = {s, t}

be a pair included in S such that m

_s,t

= m < ∞ . Then W

_X

is finite, w

_X

= (st)

^m²

if m is even, and w

_X

= (st)

^m−1²

s if m is odd. Now, let X, Y ∈ S . If m

_s,t

= 2 for all s ∈ X and t ∈ Y , then w

X

and w

Y

commute, hence (w

X

w

Y

)

²

= 1, as w

X

and w

Y

are both involutions. Suppose that (X, Y ) is a bi-orbit of type j, where j ∈ {1, 2, 3, 4, 5}.

Let Γ

₁

, . . . , Γ

_`

be the connected components of Γ

X∪Y

. For i ∈ {1, . . . , `}, we denote by Z

i

the set of vertices of Γ

i

, and we set X

i

= X ∩ Z

i

and Y

i

= Y ∩ Z

i

. We have w

X

=

Q`

i=1

w

X_i

and w

Y

=

Q`

i=1

w

Y_i

. Moreover, using the above observation together

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with Theorem

1.1, it is easily checked that (wXi

w

_Y_i

)

^m^˜^X,Y

= 1 for all i. It follows that (w

X

w

Y

)

^m^˜^X,Y

=

Q`

i=1

(w

X_i

w

Y_i

)

^m^˜^X,Y

= 1.

Lemma 3.2

Let X ∈ S . Then one of the following two alternatives holds.

(I) Γ

X

is a disjoint union of vertices (i.e. Γ

X

has no edge).

(II) There exists m ∈

N

, m ≥ 3, such that Γ

X

is a disjoint union of copies of the Coxeter graph depicted in Figure

2.1

(1).

Proof

For s ∈ X we set v

_s

(X) = |{t ∈ X | m

_s,t

≥ 3}|. Since W

_X

is finite, the connected components of Γ

X

are trees (see Bourbaki [1]), hence there exists s ∈ X such that v

s

(X) ≤ 1. On the other hand, since G acts transitively on X , we have v

_s

(X) = v

_t

(X) for all s, t ∈ X . So, either v

_s

(X) = 0 for all s ∈ X , or v

_s

(X) = 1 for all s ∈ X . If v

_s

(X) = 0 for all s ∈ X , then we are in Alternative (I). If v

_s

(X) = 1 for all s ∈ X , then we are in Alternative (II).

Let X ∈ S . We say that X is of type I if Γ

X

satisfies Condition (I) of Lemma

3.2, and

that X is of type II

m

if Γ

X

satisfies Condition (II).

Lemma 3.3

Let X, Y ∈ S , X 6= Y . Then

h ˜

X

, ˜

Y

i = − cos(π/ m ˜

_X,Y

) if m ˜

_X,Y

6= ∞ , h˜

X

, ˜

Y

i ∈ (−∞, −1] if m ˜

_X,Y

= ∞ .

Proof

Observe that, if m

_s,t

= 2 for all s ∈ X and all t ∈ Y , then h ˜

X

, ˜

Y

i = 0 and

˜

m

_X,Y

= 2. Hence, we can assume that there exist s ∈ X and t ∈ Y such that m

_s,t

≥ 3.

Since G acts transitively on X and leaves invariant Y , it follows that, for all s ∈ X , there exists t ∈ Y such that m

_s,t

≥ 3. Similarly, for all t ∈ Y , there exists s ∈ X such that m

_s,t

≥ 3.

Recall that a

_X

=

P

s∈X

s

, a

_Y

=

P

t∈Y

t

˜

X

=

_ka^a^X

Xk

, ˜

Y

=

_ka^a^Y

Yk

. Choose s ∈ X and set v

X

= |{t ∈ Y | m

_s,t

≥ 3}| and p

X

=

P

t∈Y

h

_s

,

t

i = h

_s

, a

Y

i . Since G acts transitively on X and leaves invariant Y , these definitions do not depend on the choice of s . Similarly, choose t ∈ Y and set v

_Y

= |{s ∈ X | m

_s,t

≥ 3}| and p

Y

=

P

s∈X

h

_t

,

s

i = h

_t

, a

X

i. The hypothesis that there exist s ∈ X and t ∈ Y such that m

_s,t

≥ 3 implies that v

X

≥ 1 and v

Y

≥ 1.

Let s ∈ X and t ∈ Y . If m

_s,t

≥ 3, then h

_s

,

t

i ≤ −

¹₂

, and if m

_s,t

= 2, then h

_s

,

t

i = 0.

It follows that

(3–1) p

X

≤ − v

X

2 .

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On the other hand, we have

(3–2) |X| v

X

= |Y| v

Y

.

This is the number of edges in Γ connecting an element of X with an element of Y . A direct calculation shows that

(3–3) ka

_X

k =

p

|X| if X is of type I ,

p

|X|(1 − cos(π/m)) if X is of type II

m

. Finally, by definition of p

X

,

(3–4) ha

_X

, a

_Y

i = |X| p

_X

.

Case 1: X and Y are of type I . Applying Equations (3–2), (3–3), and (3–4) we get

(3–5) h ˜

X

, ˜

Y

i = p

X

√ v

Y

√ v

_X

.

Applying Equation (3–1) to this equality we get h ˜

X

, ˜

Y

i ≤ −

√vXvY

2

. It follows that, if either v

_X

≥ 4, or v

_Y

≥ 4, or v

_X

, v

_Y

≥ 2, then h ˜

X

, ˜

Y

i ≤ −1. If v

_X

= 1, v

_Y

≥ 2 and p

X

≤ − cos(π/4) = −

^√¹

2

, then, by Equation (3–5), h ˜

X

, ˜

Y

i ≤ − 1. If v

X

= 1, v

Y

= 3 and p

_X

= − cos(π/3) = −

¹₂

, then, by Equation (3–5), h ˜

X

, ˜

Y

i = −

√ 3

2

= − cos(π/6).

In this case (Y, X) is a bi-orbit of type 5 and ˜ m

_Y,X

= m ˜

_X,Y

= 6. If v

X

= 1, v

Y

= 2 and p

_X

= − cos(π/3) = −

¹₂

, then, by Equation (3–5), h ˜

X

, ˜

Y

i = −

√ 2

2

= − cos(π/4). In this case (Y , X) is a bi-orbit of type 2 and ˜ m

_Y,X

= m ˜

_X,Y

= 4. If v

X

= 1, v

Y

= 1 and p

_X

= − cos(π/m) with m 6= ∞ , then, by Equation (3–5), h˜

X

, ˜

Y

i = − cos(π/m). In this case (Y, X) is a bi-orbit of type 1 and ˜ m

_Y,X

= m ˜

_X,Y

= m. Finally, if v

_X

= v

_Y

= 1 and p

X

≤ − 1, then, by Equation (3–5), h ˜

X

, ˜

Y

i = p

X

≤ − 1. In this case (Y, X) is a bi-orbit of type 1 and ˜ m

_Y,X

= m ˜

_X,Y

= ∞ .

Case 2: X is of type II

_m

and Y is of type I . Applying Equations (3–2), (3–3), and (3–4) we get

(3–6) h ˜

X

, ˜

Y

i = p

X

√ v

Y

p

v

X

(1 − cos(π/m)) . Applying Equation (3–1) to this equality, we get

(3–7) h ˜

X

, ˜

Y

i ≤ −

√ v

_X

v

_Y

2

p

(1 − cos(π/m)) . If m ≥ 5, then

p

1 − cos(π/m) <

¹₂

, hence, by Equation (3–7), h ˜

X

, ˜

Y

i ≤ − √ v

X

v

Y

≤

−1. So, we can assume that m ∈ {3, 4}. Then we have

p

1 − cos(π/m) ≤

^√¹

2

and, by Equation (3–7), h ˜

X

, ˜

Y

i ≤ −

√v_Xv_Y

√

2

. It follows that, if either v

X

≥ 2, or v

Y

≥ 2, then

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h ˜

X

, ˜

Y

i ≤ −1. If v

_X

= 1, v

_Y

= 1 and p

_X

≤ − cos(π/4) = −

^√¹

2

, then, by Equation (3–6), h ˜

X

, ˜

Y

i ≤ − 1. If v

X

= 1, v

Y

= 1, p

X

= − cos(π/3) = −

¹₂

and m = 4, then, by Equation (3–6), h ˜

X

, ˜

Y

i = −

√

2+√ 2

2

= − cos(π/8). In this case (Y , X) is a bi-orbit of type 4 and ˜ m

_Y,X

= m ˜

_X,Y

= 8. If v

_X

= 1, v

_Y

= 1, p

_X

= − cos(π/3) = −

¹₂

and m = 3, then, by Equation (3–6), h ˜

X

, ˜

Y

i = −

^√¹

2

= − cos(π/4). In this case (Y , X) is a bi-orbit of type 3 and ˜ m

_Y,X

= m ˜

_X,Y

= 4.

Case 3: X is of type II

m

and Y is of type II

m⁰

. Applying Equations (3–2), (3–3) and (3–4) we get

h ˜

X

, ˜

Y

i = p

X

√ v

Y

p

v

_X

(1 − cos(π/m))(1 − cos(π/m

⁰

)) . Applying Equation (3–1) to this equality we get

h˜

X

, ˜

Y

i ≤ −

√ v

X

v

Y

2

p

(1 − cos(π/m))(1 − cos(π/m

⁰

)) . Since

p

(1 − cos(π/m)) ≤

^√¹

2

and

p

(1 − cos(π/m

⁰

)) ≤

^√¹

2

, it follows that h ˜

X

, ˜

Y

i ≤

− √

v

X

v

Y

≤ −1.

Lemma 3.4

Let X ∈ S , and let x ∈ V

^G

. Then w

X

(x) = x − 2hx, ˜

X

i ˜

X

.

Proof

Let Γ

⁰

be a Coxeter graph, and let (W

⁰

, S

⁰

) be its associated Coxeter system, such that W

⁰

is finite. Let w

⁰₀

be the longest element of W

⁰

, and let (V

⁰

, h., .i

⁰

, Π

⁰

) be the canonical root basis of Γ

⁰

. Then, by Bourbaki [1], w

⁰₀

(Π

⁰

) = −Π

⁰

.

Let X ∈ S . Recall that Π

X

= {

_s

| s ∈ X}. By Lemma

2.1

and the above, we have w

X

(Π

X

) = −Π

X

, hence w

X

(a

X

) = −a

X

, therefore w

X

( ˜

X

) = −˜

X

.

Recall that V

X

denotes the linear subspace of V spanned by Π

X

. For all x ∈ V and all u ∈ W

X

there exists y ∈ V

X

such that u(x) = x + y. This is true by definition for all s ∈ X , hence it is true for all u ∈ W

_X

. Let x ∈ V

^G

. Let y ∈ V

_X

be such that w

X

(x) = x + y . Let y =

P

s∈X

λ

s

be the expression of y in the basis Π

X

. For g ∈ G we have

x +

X

s∈X

λ

s

= w

X

(x) = g(w

X

)(g(x)) = g(w

X

(x))

= g(x) +

X

s∈X

λ

s

g(

s

) = x +

X

s∈X

λ

s

_g(s)

,

hence λ

s

= λ

_g⁻¹_(s)

for all s ∈ X . Since G acts transitively on X , it follows that λ

s

= λ

t

for all s, t ∈ X . So, there exists λ ∈

R

such that w

X

(x) = x + λa

X

= x + λ ka

X

k˜

X

.

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We have

hx, ˜

X

i = hw

_X

(x), w

_X

( ˜

X

)i = hx + λ ka

_X

k ˜

X

, −˜

X

i = −hx, ˜

X

i − λ ka

_X

k , hence λ ka

_X

k = −2hx, ˜

X

i . So, w

_X

(x) = x − 2hx, ˜

X

i ˜

X

.

Proof of Theorem2.2

We have h˜

X

, ˜

X

i = 1 for all X ∈ S by definition. We have h ˜

X

, ˜

Y

i = − cos(π/ m ˜

_X,Y

) if ˜ m

_X,Y

6= ∞ ,

h˜

X

, ˜

Y

i ∈ (−∞, −1] if ˜ m

_X,Y

= ∞ ,

by Lemma

3.3. Let

χ ∈ V

^∗

be such that χ(

s

) > 0 for all s ∈ S . Let ˜ χ : V

^G

→

R

be the restriction of χ to V

^G

. Then, for X ∈ S , ˜ χ( ˜

X

) =

_ka¹

Xk

P

s∈X

χ(

s

) > 0. So, (V

^G

, h., .i, Π ˜

^G

) is a root basis of ˜ Γ .

Let ( ˜ W, S) be a Coxeter system of ˜ ˜ Γ , where ˜ S = {˜ s

_X

| X ∈ S} is a set in one- to-one correspondence with S . By Lemma

3.1, the map ˜

S → S

_W

, ˜ s

X

7→ w

X

, induces a surjective homomorphism γ : ˜ W → W

^G

. By Lemma

3.4, the composition

f

^G

◦ γ : ˜ W → GL(V

^G

) is the rooted representation associated with (V

^G

, h., .i, Π). ˜ By Theorem

1.1, it follows that

f

^G

◦ γ is injective, hence γ is an isomorphism. So, (W, S

W

) is a Coxeter system of ˜ Γ , and f

^G

: W

^G

→ GL(V

^G

) is the rooted representation associated with (V

^G

, h., .i, Π). ˜

References

[1] N Bourbaki, Eléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie.´ Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines,Actualités Scientifiques et Indus- trielles, No. 1337, Hermann, Paris, 1968.

[2] J Crisp,Symmetrical subgroups of Artin groups,Adv. Math. 152 (2000), no. 1, 159–

177.

[3] J Crisp, Erratum to: “Symmetrical subgroups of Artin groups”, Adv. Math. 179 (2003), no. 2, 318–320.

[4] J-Y Hée,Système de racines sur un anneau commutatif totalement ordonné,Geom.

Dedicata 37 (1991), no. 1, 65–102.

[5] D Krammer,The conjugacy problem for Coxeter groups,Ph. D. Thesis, Utrecht, 1994.

[6] D Krammer,The conjugacy problem for Coxeter groups,Groups Geom. Dyn. 3 (2009), no. 1, 71–171.

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[7] B M ¨uhlherr, Coxeter groups in Coxeter groups,Finite geometry and combinatorics (Deinze, 1992), 277–287, London Math. Soc. Lecture Note Ser., 191, Cambridge Univ.

Press, Cambridge, 1993.

[8] E B Vinberg,` Discrete linear groups that are generated by reflections,Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 1072–1112.

IMB, UMR 5584, CNRS, Univ. Bourgogne Franche-Comté, 21000 Dijon, France IMB, UMR 5584, CNRS, Univ. Bourgogne Franche-Comté, 21000 Dijon, France olivier.geneste@u-bourgogne.fr, lparis@u-bourgogne.fr